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<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green">E18</font></i>: 
( ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> ) &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> or 0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>suppose </b></a><a NAME="E1:7_1_1"/><i><font color="Green">E23</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:7_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:7_1_1"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:7_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:7_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E6:7_1_1"/>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E7:7_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E8:7_1_1"/>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T108">XCMPLX_1:108</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2"><b>suppose </b></a><a NAME="E1:7_1_2"/><i><font color="Green">E25</font></i>: 
0 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><a NAME="E2:7_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:7_1_2"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i/>;<br/><a NAME="E4:7_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/><a NAME="E5:7_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E6:7_1_2"/>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E7:7_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/><b>hence </b><a NAME="E8:7_1_2"/>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T108">XCMPLX_1:108</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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